Problem 47 - College Algebra and Calculus: An Applied Approach

Question

Find the area of the region bounded by the graphs of the equations. Then use a graphing utility to graph the region and verify your answer. $$ y=x^{3} e^{x}, y=0, x=0, x=2 $$

Step-by-Step Solution

Answer

The area of the region bounded by the graphs of the equations is \(8e^2 - 6\) square units.

Step 1: Identify the Function

Our curve function is \(y=x^{3} e^{x}\) and has limits \(x=0\) and \(x=2\), so we'll integrate this function within these limits to find the area under the curve.

Step 2: Set the Integral

We need to integrate the function over the interval [0, 2]. Thus, the integral we will calculate is \(\int_0^2 x^{3} e^{x} dx\)

Step 3: Perform Integration by Parts

Here it is needed to use integration by parts, where the formula is \(\int udv = uv - \int vdu\). Let \(u=x^3\), \(dv=e^{x}dx\) so that \(du=3x^2dx\) and \(v=e^{x}\). Apply these into the formula to get: \(\int_0^2 x^{3} e^{x} dx = [x^3e^x]_0^2 - \int_0^2 3x^2e^xdx\) Now we need to perform integration by parts on \(\int_0^2 3x^2e^xdx\), let \(u=x^2\), \(dv=3e^{x}dx\) so that \(du=2xdx\) and \(v=3e^{x}\). Apply these into the formula to simplify the expression: \(= [x^3e^x - 3x^2e^x]_0^2 + \int_0^2 6xe^xdx\).

Step 4: Evaluate the Integral

The remaining integral can be directly computed using the basic integration formula \(\int xe^{x}dx = (x-1)e^x + C\). So, our area is: \(A = [x^3e^x - 3x^2e^x + 6(x-1)e^x]_0^2 = (8e^2 - 6e^2 + 6e^2 - 6) - (0 - 0 + 0) = 8e^2 - 6\) square units.

Step 5: Graph Verification (optional)

Verify your answer by graphing the function \(y = x^{3}e^{x}\) from \(x = 0\) to \(x = 2\) to visualize the enclosed area. Due to the challenge of visualizing the exact area under a curve with a graphing tool, this should be used as an approximate visual confirmation, while the analytical method provides the precise numerical answer.

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus, representing the accumulation of quantities and, in geometric terms, the area under a curve between two points on the x-axis. It is symbolically represented by the integral sign with lower and upper limits, for example, \(\int_a^b f(x) dx\), where \(f(x)\) is the function to be integrated, and \(a\) and \(b\) are the limits of integration.

In the textbook exercise, to find the area under the curve described by \(y=x^{3} e^{x}\) from \(x=0\) to \(x=2\), the definite integral \(\int_0^2 x^{3} e^{x} dx\) was computed. This process effectively sums up all the infinitesimal areas under the curve from the start of the interval at \(x=0\) to the end at \(x=2\), providing the overall area encompassed between the curve, the x-axis, and the vertical lines \(x=a\) and \(x=b\).

Computing definite integrals often requires techniques such as integration by parts, substitution, or numerical methods, especially when dealing with complex functions. The steps followed in the solution illustrate how integration by parts, a method designed to handle products of functions, was used to evaluate the integral step by step and hence, find the desired area under the curve.

Area Under the Curve

Conceptual Understanding

The area under the curve is a vivid concept used to represent various physical quantities, such as distance, probability, or any quantity that accumulates over a continuous range. In this context, the curve is the graph of a function, and the area under it, between specific boundaries, is determined using definite integrals.

Application in the Exercise

The region whose area we need to find is outlined by the curve given by \(y=x^{3} e^{x}\), the x-axis (represented by \(y=0\)), and the vertical lines at \(x = 0\) and \(x = 2\). By calculating the definite integral of \(x^{3} e^{x}\) from \(0\) to \(2\), we obtain the exact area of this region.

This calculated area represents a multitude of real-life interpretations, depending on the context—including the work done by a variable force, the total growth observed over a period of time, or the cumulative probability in a distribution between two points.

Exponential Functions

Exponential functions are characterized by their base raised to a power that includes a variable, typically written as \(b^x\) or in our case, as \(e^x\), where \(e\) is the mathematical constant approximately equal to 2.71828. These functions are crucial in modeling growth and decay in disciplines such as biology, finance, physics, and many others.

Role in Integration by Parts

In the exercise, the use of an exponential function \(e^{x}\) within the integral \(\int_0^2 x^{3} e^{x} dx\) complicates the process as it grows rapidly and does not have a simple antiderivative. To handle this, integration by parts allows us to break down the product of the polynomial \(x^{3}\) and the exponential function \(e^{x}\) into simpler parts that can be integrated separately.

The presence of exponential functions in an integrand means that usual integration techniques may not directly apply, and one needs to be more strategic by using methods like integration by parts. This approach simplifies the integration process and makes it possible to evaluate integrals analytically that would otherwise be difficult or impossible to solve.